Write the equation for a parabola with a focus at $(-8,-1)$ and a directrix at $y=-4$. $y=$
Solution: The strategy A parabola is defined as the set of all points that are the same distance away from a point (the focus) and a line (the directrix). Let $(x,y)$ be a point on the parabola. Then the distance between $(x,y)$ and the focus, $(-8,-1)$, is equal to the distance between $(x,y)$ and the directrix, $y=-4$. Once we find these distances, we can equate them in order to derive the equation of our parabola. Finding the distances from $(x,y)$ to the focus and the directrix The distance between $(x,y)$ and $(-8,-1)$ is $\sqrt{(x+8)^2+(y+1)^2}$. [How did we find that?] Similarly, the distance between $(x,y)$ and the line $y=-4$ is $\sqrt{(y+4)^2}$. [How did we know that?] Deriving the formula by equating the distances $\begin{aligned} \sqrt{(y+4)^2} &= \sqrt{(x+8)^2+(y+1)^2} \\\\ (y+4)^2 &= (x+8)^2+(y+1)^2 \\\\ {y^2}+8y{+16} &= (x+8)^2{+y^2}{+2y}+1\\\\ 8y{-2y}&=(x+8)^2+1{-16} \\\\ 6y&=(x+8)^2-15 \\\\ y&=\dfrac{(x+8)^2}{6}-\dfrac{5}{2}\end{aligned}$ The answer The equation of our parabola is $y=\dfrac{(x+8)^2}{6}-\dfrac{5}{2}$. Here is the graph of our parabola. As expected, the distance between a point on the parabola, $(x,y)$, and the focus is the same as the distance between $(x,y)$ and the directrix. ${1}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${\llap{-}10}$ ${\llap{-}11}$ ${\llap{-}12}$ ${\llap{-}13}$ ${\llap{-}14}$ ${\llap{-}15}$ ${\llap{-}16}$ ${\llap{-}17}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ $y$ $x$ ${(x,y)}$